Univariate Volatility Modeling¶

Kevin Sheppard¶

Topic Outline¶

  • Visualizing Conditional Volatility
  • Testing for Conditional Volatility
  • ARCH Models
  • GARCH Models
  • Asymmetric Extensions
  • Model Building
  • Specification Checking
  • Alternative Distributional Assumptions

Data¶

S&P 500 Returns¶

In [3]:
plot(sp500)

Simulated IID $t_6$ Data¶

In [4]:
plot(iid)

FTSE 200¶

In [5]:
plot(ftse100)

GOLD Fixing Price¶

In [6]:
plot(gold)

West Texas Intermediate Crude¶

In [7]:
plot(wti)

Coinbase BitCoin¶

In [8]:
plot(btc)

Coinbase Etherium¶

In [9]:
plot(eth)

Visual Inspections for Volatility¶

  • Often sufficient to inspect return directly
  • Natural proxy for variance is $r_t^2$
    • Noisy
    • Squares distort plot (truncate at 99.5%)
  • Better proxy $|r_t|$ which is closely related to volatility

S&P 500 Squared Returns¶

In [10]:
truncated_plot(sp500 ** 2, 99.5)

$t_6$ IID Simulated Returns Squared¶

In [11]:
truncated_plot(ftse100 ** 2, 98)

West Texas Intermediate Crude Squared Returns¶

In [12]:
truncated_plot(wti ** 2, 99.5)

S&P 500 Absolute Values¶

In [13]:
truncated_plot(np.abs(sp500), 99.5)

$t_6$ IID Simulated Absolute Returns¶

In [14]:
truncated_plot(np.abs(iid), 99.9)

FTSE 100 Absolute Returns¶

In [15]:
truncated_plot(np.abs(ftse100), 99.9)

WTI Absolute Returns¶

In [16]:
truncated_plot(np.abs(wti), 99.9)

BitCoin Absolute Returns¶

In [17]:
truncated_plot(np.abs(btc), 99.9)

Formal Testing for Arch¶

  • ARCH-LM test
$$ \hat{\epsilon}_t = r_t - E_{t-1}[r_t] $$
  • Constant $E_{t-1}[r_t]=\mu$ here
$$ \hat{\epsilon}^2_t = \omega + \alpha_1 \hat{\epsilon}^2_{t-1} + \alpha_2 \hat{\epsilon}^2_{t-2} + \ldots + \alpha_p \hat{\epsilon}^2_{t-p} + \eta_t $$
  • Test statistic is $T\times R^2$
  • $\chi^2_p$ distribution
  • Virtually identical to White's test

Testing the S&P 500¶

In [19]:
arch_lm(sp500, 5)

S&P 500 between 2000 and 2009¶

In [20]:
arch_lm(sp500["2000":"2009"], 5)

Testing the S&P 500 in 2006¶

In [21]:
arch_lm(sp500["2006":"2006"], 5)

S&P 500 in 2006¶

In [22]:
plot(sp500["2006":"2006"])

BitCoin¶

In [23]:
arch_lm(btc, 5)

ARCH Models¶

  • Initial effort at volatility modeling

  • Basic ARCH(1)

$$ \begin{align*} \epsilon_t & = r_t - \mu_t \\ \sigma_t^2 & = \omega + \alpha_1 \epsilon_{t-1}^2 \\ \epsilon_t & = \sigma_t e_t \\ e_t & \stackrel{iid}{\sim} N(0,1) \\ \end{align*} $$
  • $\mu_t$ can be any mean model as long as $\in \mathcal{F}_{t-1}$
    • ARMA(P,Q)
  • Complete ARCH(P)
$$ \sigma_t^2 = \omega + \alpha_1 \epsilon_{t-1}^2 + \ldots + \alpha_P \epsilon_{t-P}^2 $$
  • In practice needs many lags

ARCH(1)¶

In [25]:
summary(arch_model(ftse100, vol="arch", p=1).fit(disp="off"))
Out[25]:
Constant Mean - ARCH Model Results
Mean Model: Constant Mean Log-Likelihood: -13702.5
Vol Model: ARCH AIC: 27411.0
Distribution: Normal BIC: 27432.4
Mean Model
coef std err t P>|t| 95.0% Conf. Int.
mu 0.0419 1.094e-02 3.825 1.306e-04 [2.041e-02,6.330e-02]
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 0.8039 3.470e-02 23.172 8.784e-119 [ 0.736, 0.872]
alpha[1] 0.3505 3.970e-02 8.828 1.066e-18 [ 0.273, 0.428]

ARCH(5)¶

In [26]:
summary(arch_model(ftse100, vol="arch", p=5).fit(disp="off"))
Out[26]:
Constant Mean - ARCH Model Results
Mean Model: Constant Mean Log-Likelihood: -12834.2
Vol Model: ARCH AIC: 25682.4
Distribution: Normal BIC: 25732.4
Mean Model
coef std err t P>|t| 95.0% Conf. Int.
mu 0.0511 9.097e-03 5.614 1.971e-08 [3.325e-02,6.891e-02]
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 0.3335 1.650e-02 20.219 6.670e-91 [ 0.301, 0.366]
alpha[1] 0.1202 1.927e-02 6.240 4.362e-10 [8.248e-02, 0.158]
alpha[2] 0.1685 1.788e-02 9.426 4.275e-21 [ 0.133, 0.204]
alpha[3] 0.1683 2.238e-02 7.519 5.504e-14 [ 0.124, 0.212]
alpha[4] 0.1429 1.717e-02 8.319 8.888e-17 [ 0.109, 0.177]
alpha[5] 0.1314 2.069e-02 6.349 2.173e-10 [9.082e-02, 0.172]

ARCH(15)¶

In [28]:
plot_arch(15)

HARCH Model¶

  • Compact parameterization of long-lag ARCH
  • Define
$$ \overline{r^2_{t-1:j}} = j^{-1} \sum_{i=1}^j r_{t-i}^2 $$
  • HARCH is parameterized in terms of $\overline{r^2_{t-1:j}}$
  • Standard model: HARCH(1,5,22)
$$ \sigma^2_t = \omega + \alpha_1 r_{t-1}^2 + \alpha_5 \overline{r^2_{t-1:5}} + \alpha_{22} \overline{r^2_{t-1:22}} $$
  • Compute effective coefficient by expanding

HARCH(1, 5, 22) Model¶

In [29]:
summary(arch_model(ftse100, vol="harch",p=[1,5,22]).fit(disp="off"))
Out[29]:
Constant Mean - HARCH Model Results
Mean Model: Constant Mean Log-Likelihood: -12679.1
Vol Model: HARCH AIC: 25368.1
Distribution: Normal BIC: 25403.9
Mean Model
coef std err t P>|t| 95.0% Conf. Int.
mu 0.0459 8.614e-03 5.329 9.880e-08 [2.902e-02,6.278e-02]
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 0.1870 1.860e-02 10.053 8.955e-24 [ 0.151, 0.223]
alpha[1] 8.5708e-15 2.450e-02 3.498e-13 1.000 [-4.802e-02,4.802e-02]
alpha[5] 0.3609 6.262e-02 5.764 8.232e-09 [ 0.238, 0.484]
alpha[22] 0.4859 4.299e-02 11.302 1.281e-29 [ 0.402, 0.570]

Extended HARCH Model¶

In [30]:
summary(arch_model(ftse100, vol="harch",p=[1,5,22, 66]).fit(disp="off"))
Out[30]:
Constant Mean - HARCH Model Results
Mean Model: Constant Mean Log-Likelihood: -12654.9
Vol Model: HARCH AIC: 25321.8
Distribution: Normal BIC: 25364.8
Mean Model
coef std err t P>|t| 95.0% Conf. Int.
mu 0.0473 8.620e-03 5.487 4.081e-08 [3.041e-02,6.420e-02]
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 0.1482 1.930e-02 7.678 1.615e-14 [ 0.110, 0.186]
alpha[1] 1.0355e-03 2.369e-02 4.372e-02 0.965 [-4.539e-02,4.746e-02]
alpha[5] 0.3830 6.023e-02 6.358 2.037e-10 [ 0.265, 0.501]
alpha[22] 0.2765 6.112e-02 4.524 6.058e-06 [ 0.157, 0.396]
alpha[66] 0.2122 4.437e-02 4.781 1.740e-06 [ 0.125, 0.299]

HARCH(1,5,22,66) expanded coefficients¶

In [32]:
harch_weights()

The GARCH Model¶

  • Simple but effective generalization of ARCH
$$ \sigma^2_t = \omega + \alpha_1 \epsilon_{t-1}^2 + \beta_1 \sigma^2_{t-1} $$
  • ARCH($\infty$) in disguise
$$ \sigma_t^2 = \frac{\omega}{1-\beta} + \alpha_1 \sum_{i=1}^\infty \beta_1^{i-1} \epsilon_{t-i}^2 $$
  • Similar to ARMA(1,1)
    • $\alpha+\beta$ is the AR coefficient
    • $-\beta$ is the MA coefficient

GARCH Model Results¶

In [33]:
garch = arch_model(ftse100, vol="garch").fit(disp="off")
summary(garch)
Out[33]:
Constant Mean - GARCH Model Results
Mean Model: Constant Mean Log-Likelihood: -12665.8
Vol Model: GARCH AIC: 25339.7
Distribution: Normal BIC: 25368.3
Mean Model
coef std err t P>|t| 95.0% Conf. Int.
mu 0.0483 8.961e-03 5.395 6.854e-08 [3.078e-02,6.591e-02]
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 0.0203 4.160e-03 4.868 1.126e-06 [1.210e-02,2.840e-02]
alpha[1] 0.1008 1.198e-02 8.418 3.821e-17 [7.736e-02, 0.124]
beta[1] 0.8824 1.324e-02 66.656 0.000 [ 0.856, 0.908]

The Conditional Mean¶

  • All models have used a constant mean $\mu_t = \mu$
  • Plausible for liquid financial assets
  • For high-frequency data, mean makes little difference
  • Can instead use $\mu_t = 0$

Zero mean¶

In [34]:
zero = arch_model(ftse100, vol="garch", mean="zero").fit(disp="off")
summary(zero)
Out[34]:
Zero Mean - GARCH Model Results
Mean Model: Zero Mean Log-Likelihood: -12682.3
Vol Model: GARCH AIC: 25370.7
Distribution: Normal BIC: 25392.1
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 0.0196 3.966e-03 4.929 8.244e-07 [1.178e-02,2.733e-02]
alpha[1] 0.0977 1.111e-02 8.793 1.456e-18 [7.589e-02, 0.119]
beta[1] 0.8861 1.250e-02 70.883 0.000 [ 0.862, 0.911]

Zero vs. Non-zero Mean¶

In [36]:
compare_mean_models()

Exponentially Weighted Moving Average (EWMA) Variance¶

  • Restricted version of GARCH
$$ \begin{align*} \omega & = 0 \\ \alpha+\beta& =1 \Rightarrow \alpha = 1-\beta \end{align*} $$
  • Usually parameterize with $\lambda$
$$ \sigma_t^2 = (1-\lambda)\epsilon_{t-1}^2 + \lambda \sigma^2_{t-1} $$
  • Simpler if $\mu_t=0$
$$ \sigma_t^2 = (1-\lambda)r_{t-1}^2 + \lambda \sigma^2_{t-1} $$
  • Common to fix $\lambda$
    • Daily: 0.94
    • Weekly: 0.97
    • Monthly: 0.99

EWMA Results¶

In [37]:
ewma_var = arch.univariate.volatility.EWMAVariance(0.94)
summary(arch.univariate.ConstantMean(ftse100, volatility=ewma_var).fit(disp="off"))
Out[37]:
Constant Mean - EWMA/RiskMetrics Model Results
Mean Model: Constant Mean Log-Likelihood: -12757.7
Vol Model: EWMA/RiskMetrics AIC: 25517.4
Distribution: Normal BIC: 25524.6
Mean Model
coef std err t P>|t| 95.0% Conf. Int.
mu 0.0426 1.015e-02 4.196 2.721e-05 [2.269e-02,6.247e-02]

Asymmetries in Conditional Variance¶

GJR-GARCH¶

  • Adds asymmetry to GARCH model
$$ \sigma^2_t = \omega + \alpha_1 \epsilon_{t-1}^2 + \gamma_1 I_{[\epsilon_t-1<0]} \epsilon_{t-1}^2 + \beta_1 \sigma^2_{t-1} $$

GJR-GARCH(1,1,) Model¶

In [38]:
summary(arch_model(ftse100, vol="garch", o=1).fit(disp="off"))
Out[38]:
Constant Mean - GJR-GARCH Model Results
Mean Model: Constant Mean Log-Likelihood: -12574.3
Vol Model: GJR-GARCH AIC: 25158.6
Distribution: Normal BIC: 25194.4
Mean Model
coef std err t P>|t| 95.0% Conf. Int.
mu 0.0245 8.442e-03 2.907 3.646e-03 [7.997e-03,4.109e-02]
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 0.0207 3.938e-03 5.262 1.424e-07 [1.300e-02,2.844e-02]
alpha[1] 0.0263 6.720e-03 3.910 9.234e-05 [1.310e-02,3.945e-02]
gamma[1] 0.1126 1.802e-02 6.246 4.216e-10 [7.724e-02, 0.148]
beta[1] 0.8973 1.184e-02 75.758 0.000 [ 0.874, 0.921]

The TARCH Model¶

  • Changes model from variance to volatility
$$ \sigma_t = \omega + \alpha_1 |\epsilon_{t-1}| + \gamma_1 I_{[\epsilon_t-1<0]} |\epsilon_{t-1}| + \beta_1 \sigma_{t-1} $$

TARCH(1,1,1)¶

In [39]:
summary(arch_model(ftse100, vol="garch", o=1, power=1).fit(disp="off"))
Out[39]:
Constant Mean - TARCH/ZARCH Model Results
Mean Model: Constant Mean Log-Likelihood: -12566.9
Vol Model: TARCH/ZARCH AIC: 25143.9
Distribution: Normal BIC: 25179.6
Mean Model
coef std err t P>|t| 95.0% Conf. Int.
mu 0.0204 8.517e-03 2.401 1.636e-02 [3.754e-03,3.714e-02]
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 0.0230 4.398e-03 5.223 1.763e-07 [1.435e-02,3.159e-02]
alpha[1] 0.0409 7.306e-03 5.602 2.124e-08 [2.660e-02,5.524e-02]
gamma[1] 0.0980 1.332e-02 7.359 1.850e-13 [7.192e-02, 0.124]
beta[1] 0.9070 1.098e-02 82.584 0.000 [ 0.885, 0.929]

Exponential GARCH¶

$$ \ln \sigma^2_t = \omega + \alpha_1 \left| \frac{\epsilon_{t-1}}{\sigma_{t-1}} - \sqrt{\frac{2}{\pi}} \right| + \gamma_1 \frac{\epsilon_{t-1}}{\sigma_{t-1}} + \beta_1 \ln \sigma^2_{t-1} $$

EGARCH(1,1,1)¶

In [40]:
summary(arch_model(ftse100, vol="egarch", o=1).fit(disp="off"))
Out[40]:
Constant Mean - EGARCH Model Results
Mean Model: Constant Mean Log-Likelihood: -12565.9
Vol Model: EGARCH AIC: 25141.8
Distribution: Normal BIC: 25177.5
Mean Model
coef std err t P>|t| 95.0% Conf. Int.
mu 0.0214 5.552e-03 3.861 1.128e-04 [1.056e-02,3.232e-02]
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 7.3006e-04 1.948e-03 0.375 0.708 [-3.089e-03,4.549e-03]
alpha[1] 0.1605 1.612e-02 9.953 2.447e-23 [ 0.129, 0.192]
gamma[1] -0.0817 9.921e-03 -8.238 1.748e-16 [ -0.101,-6.229e-02]
beta[1] 0.9794 3.723e-03 263.034 0.000 [ 0.972, 0.987]

Asymmetric Power ARCH¶

  • Parameterizes power in model
$$ \sigma^\delta_t = \omega + \alpha_1 \left(|\epsilon_{t-1}| + \gamma_1 \epsilon_{t-1}\right)^\delta + \beta_1 \sigma^\delta_{t-1} $$

APARCH(1,1,1) Model¶

In [41]:
from arch.univariate import ConstantMean, APARCH
summary(ConstantMean(ftse100, volatility=APARCH()).fit(disp="off"))

C:\Anaconda\lib\site-packages\scipy\optimize\optimize.py:282: RuntimeWarning: Values in x were outside bounds during a minimize step, clipping to bounds
  warnings.warn("Values in x were outside bounds during a "
Out[41]:
Constant Mean - APARCH Model Results
Mean Model: Constant Mean Log-Likelihood: -12559.6
Vol Model: APARCH AIC: 25131.3
Distribution: Normal BIC: 25174.2
Mean Model
coef std err t P>|t| 95.0% Conf. Int.
mu 0.0216 8.536e-03 2.527 1.151e-02 [4.838e-03,3.830e-02]
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 0.0220 4.132e-03 5.325 1.009e-07 [1.391e-02,3.010e-02]
alpha[1] 0.0854 9.696e-03 8.805 1.314e-18 [6.637e-02, 0.104]
gamma[1] 0.4937 6.798e-02 7.262 3.816e-13 [ 0.360, 0.627]
beta[1] 0.9048 1.096e-02 82.593 0.000 [ 0.883, 0.926]
delta 1.3226 0.158 8.384 5.124e-17 [ 1.013, 1.632]

ACF and PACF of $\epsilon^{2}_{t}$¶

  • Time-varying volatility appears through persistence in $\epsilon^{2}_{t}$
    • Square is essential
  • Standard tool is to plot ACF and PACF of $\hat{\epsilon}^{2}_{t}$
    • Estimate mean without time-varying volatility
  • ACF indicates ARCH-like terms needed
  • PACF indicates GARCH-like terms needed

S&P 500¶

In [43]:
acf_pacf_plot(sp500**2, 22)

West Texas Internediate Crude¶

In [44]:
acf_pacf_plot(wti**2, 22)

Bitcoin¶

In [45]:
acf_pacf_plot(btc**2, 22)

Etherium¶

In [46]:
acf_pacf_plot(eth**2, 22)

Model Building¶

  • Usually begin with GARCH
  • Check lag specification using StG
  • Check for asymmetries using GJR-GARCH
  • Check for functional form using TARCH and EGARCH
In [47]:
garch11 = arch_model(eth).fit(disp="off")
summary(garch11)
Out[47]:
Constant Mean - GARCH Model Results
Mean Model: Constant Mean Log-Likelihood: -5315.65
Vol Model: GARCH AIC: 10639.3
Distribution: Normal BIC: 10661.1
Mean Model
coef std err t P>|t| 95.0% Conf. Int.
mu 0.2984 0.113 2.643 8.227e-03 [7.709e-02, 0.520]
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 3.3878 1.542 2.197 2.804e-02 [ 0.365, 6.410]
alpha[1] 0.1739 4.966e-02 3.503 4.606e-04 [7.661e-02, 0.271]
beta[1] 0.7337 8.177e-02 8.973 2.897e-19 [ 0.573, 0.894]
In [48]:
garch21 = arch_model(eth, p=2).fit(disp="off")
summary(garch21)
Out[48]:
Constant Mean - GARCH Model Results
Mean Model: Constant Mean Log-Likelihood: -5315.65
Vol Model: GARCH AIC: 10641.3
Distribution: Normal BIC: 10668.6
Mean Model
coef std err t P>|t| 95.0% Conf. Int.
mu 0.2984 0.114 2.623 8.708e-03 [7.546e-02, 0.521]
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 3.3876 1.974 1.716 8.613e-02 [ -0.481, 7.257]
alpha[1] 0.1740 5.461e-02 3.185 1.445e-03 [6.692e-02, 0.281]
alpha[2] 8.3744e-16 7.523e-02 1.113e-14 1.000 [ -0.147, 0.147]
beta[1] 0.7337 0.114 6.441 1.185e-10 [ 0.510, 0.957]
In [49]:
garch12 = arch_model(eth, q=2).fit(disp="off")
summary(garch12)
Out[49]:
Constant Mean - GARCH Model Results
Mean Model: Constant Mean Log-Likelihood: -5312.01
Vol Model: GARCH AIC: 10634.0
Distribution: Normal BIC: 10661.3
Mean Model
coef std err t P>|t| 95.0% Conf. Int.
mu 0.2985 0.112 2.657 7.890e-03 [7.828e-02, 0.519]
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 3.8957 1.679 2.320 2.036e-02 [ 0.604, 7.187]
alpha[1] 0.2135 5.746e-02 3.715 2.031e-04 [ 0.101, 0.326]
beta[1] 0.4137 0.109 3.812 1.381e-04 [ 0.201, 0.626]
beta[2] 0.2677 8.408e-02 3.184 1.451e-03 [ 0.103, 0.433]
In [50]:
garch22 = arch_model(eth, p=2, q=2).fit(disp="off")
summary(garch22)
Out[50]:
Constant Mean - GARCH Model Results
Mean Model: Constant Mean Log-Likelihood: -5312.01
Vol Model: GARCH AIC: 10636.0
Distribution: Normal BIC: 10668.7
Mean Model
coef std err t P>|t| 95.0% Conf. Int.
mu 0.2985 0.113 2.644 8.186e-03 [7.724e-02, 0.520]
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 3.8950 3.776 1.031 0.302 [ -3.506, 11.296]
alpha[1] 0.2135 7.039e-02 3.033 2.422e-03 [7.552e-02, 0.351]
alpha[2] 2.6050e-15 0.190 1.369e-14 1.000 [ -0.373, 0.373]
beta[1] 0.4137 0.530 0.780 0.435 [ -0.625, 1.453]
beta[2] 0.2678 0.286 0.935 0.350 [ -0.294, 0.829]
In [51]:
gjrgarch112 = arch_model(eth, q=2, o=1).fit(disp="off")
summary(gjrgarch112)
Out[51]:
Constant Mean - GJR-GARCH Model Results
Mean Model: Constant Mean Log-Likelihood: -5311.95
Vol Model: GJR-GARCH AIC: 10635.9
Distribution: Normal BIC: 10668.6
Mean Model
coef std err t P>|t| 95.0% Conf. Int.
mu 0.2892 0.115 2.520 1.174e-02 [6.427e-02, 0.514]
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 3.9218 1.660 2.362 1.818e-02 [ 0.667, 7.176]
alpha[1] 0.2087 6.364e-02 3.279 1.041e-03 [8.396e-02, 0.333]
gamma[1] 0.0142 7.067e-02 0.201 0.841 [ -0.124, 0.153]
beta[1] 0.4136 0.108 3.827 1.295e-04 [ 0.202, 0.625]
beta[2] 0.2657 8.858e-02 3.000 2.703e-03 [9.210e-02, 0.439]
In [52]:
tarch112 = arch_model(eth, q=2, o=1, power=1.0).fit(disp="off")
summary(tarch112)
Out[52]:
Constant Mean - TARCH/ZARCH Model Results
Mean Model: Constant Mean Log-Likelihood: -5311.64
Vol Model: TARCH/ZARCH AIC: 10635.3
Distribution: Normal BIC: 10668.0
Mean Model
coef std err t P>|t| 95.0% Conf. Int.
mu 0.3063 0.128 2.401 1.634e-02 [5.629e-02, 0.556]
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 0.5484 0.247 2.216 2.671e-02 [6.330e-02, 1.033]
alpha[1] 0.1965 4.514e-02 4.353 1.344e-05 [ 0.108, 0.285]
gamma[1] -4.9817e-03 4.319e-02 -0.115 0.908 [-8.964e-02,7.967e-02]
beta[1] 0.4471 9.875e-02 4.528 5.958e-06 [ 0.254, 0.641]
beta[2] 0.3178 9.509e-02 3.342 8.324e-04 [ 0.131, 0.504]
In [53]:
egarch112 = arch_model(eth, vol="EGARCH", q=2).fit(disp="off")
summary(egarch112)
Out[53]:
Constant Mean - EGARCH Model Results
Mean Model: Constant Mean Log-Likelihood: -5308.34
Vol Model: EGARCH AIC: 10626.7
Distribution: Normal BIC: 10653.9
Mean Model
coef std err t P>|t| 95.0% Conf. Int.
mu 0.3003 0.121 2.482 1.305e-02 [6.319e-02, 0.537]
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 0.3951 0.149 2.658 7.863e-03 [ 0.104, 0.686]
alpha[1] 0.3671 7.099e-02 5.171 2.324e-07 [ 0.228, 0.506]
beta[1] 0.5440 8.495e-02 6.404 1.514e-10 [ 0.377, 0.710]
beta[2] 0.3488 8.511e-02 4.098 4.169e-05 [ 0.182, 0.516]
In [54]:
egarch102 = arch_model(eth, vol="EGARCH", o=0, q=2).fit(disp="off")
summary(egarch102)
Out[54]:
Constant Mean - EGARCH Model Results
Mean Model: Constant Mean Log-Likelihood: -5308.34
Vol Model: EGARCH AIC: 10626.7
Distribution: Normal BIC: 10653.9
Mean Model
coef std err t P>|t| 95.0% Conf. Int.
mu 0.3003 0.121 2.482 1.305e-02 [6.319e-02, 0.537]
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 0.3951 0.149 2.658 7.863e-03 [ 0.104, 0.686]
alpha[1] 0.3671 7.099e-02 5.171 2.324e-07 [ 0.228, 0.506]
beta[1] 0.5440 8.495e-02 6.404 1.514e-10 [ 0.377, 0.710]
beta[2] 0.3488 8.511e-02 4.098 4.169e-05 [ 0.182, 0.516]

Summary¶

In [56]:
models()
Out[56]:
omega alpha[1] alpha[2] gamma[1] beta[1] beta[2] LLF AIC BIC
GARCH(1,1) 3.388 0.174 0.734 -5315.7 10639.3 10661.1
2.197 3.503 8.973
GARCH(1,2) 3.896 0.213 0.414 0.268 -5312.0 10634.0 10661.3
2.320 3.715 3.812 3.184
GARCH(2,1) 3.388 0.174 0.000 0.734 -5315.7 10641.3 10668.6
1.716 3.185 0.000 6.441
GARCH(2,2) 3.895 0.213 0.000 0.414 0.268 -5312.0 10636.0 10668.7
1.031 3.033 0.000 0.780 0.935
GJR(1,1,2) 3.922 0.209 0.014 0.414 0.266 -5311.9 10635.9 10668.6
2.362 3.279 0.201 3.827 3.000
TARCH(1,1,2) 0.548 0.196 -0.005 0.447 0.318 -5311.6 10635.3 10668.0
2.216 4.353 -0.115 4.528 3.342
EGARCH(1,1,2) 0.395 0.367 0.544 0.349 -5308.3 10626.7 10653.9
2.658 5.171 6.404 4.098
EGARCH(1,0,2) 0.395 0.367 0.544 0.349 -5308.3 10626.7 10653.9
2.658 5.171 6.404 4.098

Specification Checking¶

  • Conceptually similar to mean modeling
  • Emphasis of different form of residual
$$ \hat{e}_t^2 = \frac{\hat{\epsilon}_t^2}{\hat{\sigma}_t^2}$$
  • Check ACF/PACF for White Noise appearance

ACF and PACF of $\hat{e}^2_t$¶

In [57]:
acf_pacf_plot(egarch102.std_resid ** 2, 22)

ARCH-LM test¶

In [58]:
arch_lm(egarch102.std_resid, 10)

Alternative Distributional Assumptions¶

  • Have assumed $e_t \stackrel{iid}{\sim} N(0,1)$
  • Can relax this assumption
  • Common alternatives:
    • Student's $t$
    • Generalized Error Distribution
    • Hansen's Skew $t$

Baseline Normal¶

In [59]:
normal = arch_model(sp500, o=1, vol="egarch").fit(disp="off")
summary(normal, mean=False)
Out[59]:
Constant Mean - EGARCH Model Results
Mean Model: Constant Mean Log-Likelihood: -13681.8
Vol Model: EGARCH AIC: 27373.6
Distribution: Normal BIC: 27409.8
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 2.8410e-03 2.112e-03 1.345 0.179 [-1.299e-03,6.981e-03]
alpha[1] 0.1587 1.692e-02 9.380 6.584e-21 [ 0.126, 0.192]
gamma[1] -0.1040 1.256e-02 -8.280 1.229e-16 [ -0.129,-7.935e-02]
beta[1] 0.9757 4.241e-03 230.054 0.000 [ 0.967, 0.984]

Standardized Students's $t$¶

  • Student's $t$ normalized to always have variance 1
  • $\nu$ controls tail thickness
In [60]:
studt = arch_model(sp500, o=1, vol="egarch", dist="t").fit(disp="off")
summary(studt, mean=False)
Out[60]:
Constant Mean - EGARCH Model Results
Mean Model: Constant Mean Log-Likelihood: -13408.6
Vol Model: EGARCH AIC: 26829.2
Distribution: Standardized Student's t BIC: 26872.7
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 1.8400e-03 1.820e-03 1.011 0.312 [-1.728e-03,5.407e-03]
alpha[1] 0.1489 1.025e-02 14.524 8.578e-48 [ 0.129, 0.169]
gamma[1] -0.1040 8.646e-03 -12.027 2.562e-33 [ -0.121,-8.704e-02]
beta[1] 0.9825 2.586e-03 379.952 0.000 [ 0.977, 0.988]
Distribution
coef std err t P>|t| 95.0% Conf. Int.
nu 6.4775 0.436 14.865 5.549e-50 [ 5.623, 7.332]

Generalized Error Distribution¶

  • Nests the normal when $\nu=2$
  • Heavy tailed for $\nu<2$
In [61]:
ged = arch_model(sp500, o=1, vol="egarch", dist="ged").fit(disp="off")
summary(ged, mean=False)
Out[61]:
Constant Mean - EGARCH Model Results
Mean Model: Constant Mean Log-Likelihood: -13426.3
Vol Model: EGARCH AIC: 26864.6
Distribution: Generalized Error Distribution BIC: 26908.0
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 1.3248e-03 1.841e-03 0.720 0.472 [-2.284e-03,4.933e-03]
alpha[1] 0.1514 1.178e-02 12.848 8.805e-38 [ 0.128, 0.174]
gamma[1] -0.1025 9.533e-03 -10.752 5.811e-27 [ -0.121,-8.381e-02]
beta[1] 0.9805 2.975e-03 329.522 0.000 [ 0.975, 0.986]
Distribution
coef std err t P>|t| 95.0% Conf. Int.
nu 1.3358 3.350e-02 39.879 0.000 [ 1.270, 1.401]

Hansen's Skew $t$¶

  • Extendeds the $t$ with a skewness parameter $\lambda$
In [62]:
skewt = arch_model(sp500, o=1, vol="egarch", dist="skewt").fit(disp="off")
summary(skewt, mean=False)
Out[62]:
Constant Mean - EGARCH Model Results
Mean Model: Constant Mean Log-Likelihood: -13394.4
Vol Model: EGARCH AIC: 26802.9
Distribution: Standardized Skew Student's t BIC: 26853.6
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 3.1877e-03 1.928e-03 1.653 9.825e-02 [-5.911e-04,6.966e-03]
alpha[1] 0.1494 1.012e-02 14.757 2.755e-49 [ 0.130, 0.169]
gamma[1] -0.1058 8.618e-03 -12.271 1.296e-34 [ -0.123,-8.886e-02]
beta[1] 0.9815 2.622e-03 374.307 0.000 [ 0.976, 0.987]
Distribution
coef std err t P>|t| 95.0% Conf. Int.
nu 6.6500 0.452 14.718 4.944e-49 [ 5.764, 7.536]
lambda -0.0705 1.241e-02 -5.683 1.323e-08 [-9.487e-02,-4.621e-02]

Comparing Volatility¶

In [64]:
compare_volatility(annualize=True)

Comparing Volatility in 2020¶

In [65]:
compare_volatility(annualize=True, span="2020")

Comparing the estimated distributions¶

In [68]:
compare_dist()

Comparing the Tails¶

In [69]:
compare_dist(tails=True)

Next Week¶

Univariate Volatility Modeling¶

  • Forecasting
  • Forecast Evaluation
  • Realized Variance

Happy New Year¶

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